Physical systems can be modeled mathematically to simulate their behavior under certain conditions. There are a wide variety of means to model these systems, ranging from the very simplistic to the extremely complicated. One of the more complicated means to model physical systems is through the use of finite element analysis. As the name implies, finite element analysis involves the representation of individual, finite elements of the physical system in a mathematical model and the solution of this model in the presence of a predetermined set of boundary conditions.
In finite element modeling, the region that is to be analyzed is broken up into sub-regions called elements. This process of dividing the region into sub-regions may be referred to as discretization or mesh generation. The region is represented by functions defined over each element. This generates a number of local functions that are much simpler than those which would be required to represent the entire region. The next step is to analyze the response for each element. This is accomplished by building a matrix that defines the properties of the various elements within the region and a vector that defines the forces acting on each element in the structure. Once all the element matrices and vectors have been created, they are combined into a structure matrix equation. This equation relates nodal responses for the entire structure to nodal forces. After applying boundary conditions, the structure matrix equation can be solved to obtain unknown nodal responses. Intra-element responses can be interpolated from nodal values using the functions which were defined over each element.
In prior art methods, when a solution to a structured three-dimensional model over time was desired, the model had be solved for a first time value, then again for a second time value, and so on, until a series of time steps covering the desired range was completed. If it was desired to model a temporal feature more accurately, the size of the time steps had to be decreased and the entire three-dimensional physical model had to be solved for a greater number of time values. The computing resources which were required to generate the solution obviously increased with the number of time steps for which the model was solved. There was no way to localize the smaller time steps to a particular physical volume of interest or to adapt the three-dimensional spatial model over time. Further, when an error occurred in the solution of the model for one of the time values, the model and/or the time steps had to be adjusted and the entire model solved again for each of the time values.